This document is a math studies internal assessment that investigates the relationship between SAT scores and family income of test takers around the world. The student analyzed data on SAT scores and family incomes from the College Board in 2007. A scatter plot, least squares regression line, and correlation coefficient calculation showed a strong positive correlation between higher SAT scores and higher family incomes. A chi-squared test rejected the null hypothesis that SAT scores and family income are independent. However, limitations in the data are noted, such as incomplete income reporting and wide income brackets in the raw data.

1. IB Math Studies Internal Assessment:
What is the Relationship between SAT Scores and Family Income of
the Test Takers around the World?
Exam Session: May 2011
School name: Allen High School
Teacher: Mr. Arnold-Roksandich
Date: January 22nd, 2014
Course: IB Math Studies
Word Count: 1,832
Name: Travis Hayes

2. What is the Relationship between SAT Scores and Family Income of the Test
Takers around the World?
Introduction
The SAT examination is mostly in today’s world of academics, a
requirement of getting accepted into collage. Not only is it enough to take the
examination but the student has to pass with an average score or above to even
have his/her application be considered. Many students around the world
recognize this and therefore apply to prep schools for the SAT or their parents
send them to a higher educational institution for that purpose. The prep schools
such as Princeton are not cheap however as it helps give advice on how to best
tackle the SAT examination, neither are higher educational institutions. Also it
can be considered a luxury service by some middle class and low class societies
in the world to be able to attend either one. This being said, the SAT prep course
and higher educational institutions are,as a result, aimed at the high class
societies in the world or those who can afford it. If this is true, it would putfamilies
with a higher income at an advantage for their childrento get accepted into
collage compared to families who cannot afford for their children to take the
course or school fee and learn the advice of how to pass the SAT examination
with a high score.Are the collageswhich students aim to be accepted into for a
better education really based on which families can afford for their children to
take the SAT prep course or learn at a higher educational institution? The data
collected from Collage Board in year2007 was analyzed to determine whether
there is a relationship between SAT scores and family income of the test takers
around the world (Rampell).
Statement of Task
The main purpose of this investigation is to determine whether there is a
relationship between SAT scores and family income of the test takers around the
world. The type of data that will be collected is the SAT scores and family income
of the two-thirds of test takers who voluntarily reported it to collage board when
signing up for the SAT examination worldwide.The SAT scores are used to
determine how high of a score the test taker got and family income to determine
the possibility to send their children to SAT prep schools or better educational
institutions.The data used to generate the data breaks down the average score
for ten different income groups of $20,000 range.

3. Plan of investigation
I am investigating the relationship of SAT scores and family income of the test
takers around the world. I have collected data on SAT scores and family income of the
test takers around the world. With the collection of data that I have acquired, a number
of mathematical processes were used to analyze the data: a scatter plot of the data,
calculation of the least squares regression line and correlation coefficient. I am going to
do a χ2 test on the data to show the dependence of SAT scores and family income of the
test takers around the world.

4. Mathematical Investigation
Collected Data
Table 1: Mean SAT scores per section categorized in family income of test taker
in 2007
Family income of test
takers
Percentage of test takers
within each family income
group
Critical
reading
Math
Writing
Less than $10,000
4%
427
451
423
1301
$10,000–$20,000
8%
453
472
446
1371
$20,000–$30,000
6%
454
465
444
1363
$30,000–$40,000
9%
476
485
466
1427
$40,000–$50,000
8%
489
496
477
1462
$50,000–$60,000
8%
497
504
486
1487
$60,000–$70,000
8%
504
511
493
1508
$70,000–$80,000
9%
508
516
498
1522
$80,000–$100,000
14%
520
529
510
1559
More than $100,000
26%
544
556
537
1637
This bottom row, the “More than $100,000” I am going to consider as an
outlier therefore excluded in all calculations as it goes from $100,000 up to the
millions of dollar of income which is too wide of a range to include into the
calculations of this assessment.

5. Graph 1: Average SAT Score Vs. Family Income
1600
Overall Averaged SAT Score (top score 2400)
1559
1550
1508
1522
1487
1500
1462
1450
1427
1400
1371
Average SAT score
1363
1350
1301
1300
1250
0
20
40
60
80
100
Family Income of SAT Takers ($ in Thousands)
Graph 1 shows the average SAT score Vs. family income of test taker. As of now,
there seems to be very strong positive correlation. It does appear that the SAT
scores improve as the family income increases. (Graph was generated through
Microsoft Excel)

6. Calculation of the Least Squares Regression
The Least Square regression identifies the relationship between the
independent variable, x, and the dependent variable, y. It is given by the
following formula:
where
and
Table 2: Values of Least Squares Regression
x
y
xy
x2
15000
1301
19515000
225000000
25000
1371
34275000
625000000
35000
1363
47705000
1225000000
45000
1427
64215000
2025000000
55000
1462
80410000
3025000000
65000
1487
96655000
4225000000
75000
1508
113100000
5625000000
85000
1522
129370000
7225000000
95000
1559
148105000
9025000000
∑ = 495000
∑ = 13000
∑ = 733350000
∑ = 33225000000
= 55000
= 79444444.44
= 1444.
= 3691666667
These are the calculated values used in finding the Least Squares Regression
2

7. Calculation of Pearson’s Correlation Coefficient
Pearson’s Correlation Coefficient indicates the strength of the relationship
between the two variables (SAT scores and family income of test taker). It is
given by the following formula:
where
,
and
Table 3: Values of Pearson’s Correlation Coefficient
x
y
15000
1301
1600000000
20576.30864
25000
1371
900000000
5394.08642
35000
1363
400000000
6633.197531
45000
1427
100000000
304.308642
1462
55000
0
308.1975309
65000
1487
100000000
1810.975309
75000
1508
400000000
4039.308642
85000
1522
900000000
6014.864198
95000
1559
1600000000
13122.97531
∑ = 495000
∑ = 13000
∑ = 6000000000
∑ = 58204.22222
= 55000
= 1444.
These are the calculated values used in finding the Correlation Coefficient.
0.9819360378

8. The calculation
suggests that the strength of the
association of the data is very strong since 0.90 r2<1.
I compared this value of
with the standard table of coefficient of
determinations which places it in the “very strong” category (Whiffen).
Overall Averaged SAT Score (top score 2400)
Graph 2: Average SAT Score Vs. Family Income
Linear Fit line
1600
1559
1550
1508
1522
1487
1500
1462
1427
1450
1400
Average SAT score
1371 1363
Linear (Average SAT score)
1350
1301
1300
1250
0
20
40
60
80
100
Family Income of SAT Takers ($ in Thousands)
Graph 2 indicates that there is a strong positive linear correlation. This is also
indicated through the value of correlation coefficient, 0.96.(the graph was generated
through Microsoft Excel )
Calculation of a
2
test
The 2 test is used to measure whether two classifications or factors from
the same sample are independent of each other– if the occurrence of one of
them does not affect the occurrence of the other.

9. Observed Values:
A1
A2
Total
B1
A
C
A+C
B2
B
D
B+D
Total
A+B
C+D
N
B2
Total
Calculations of Expected Values:
B1
A1
A+B
A2
C+D
Total
A+C
B+D
N
Degrees of freedom measure the number of values in the final calculation that
are free to vary:
Null (H0) Hypothesis: SAT scores and family income are independent from each
other.
Alternative (H1) Hypothesis: SAT scores and family income are dependent from
each other.

10. Table 4: Observation Values
Score
Income($)
1300-1430
1431-1561
Total
15000 – 55000
4
1
5
56000 – 96000
4
4
Total
4
5
9
Table 2 shows the observed values for SAT score Vs. family income. The data
pieces have been put into ranges that represent the income of the families of the
test takers.
Table 5: Calculations for the Expected Values
Income($)
1300-1430
Score
1300-1430
Total
15000 – 55000
4+1
56000 – 96000
0+4
Total
4+0
1+4
9
Table 3 shows the individual calculations for each of the expected values.
Table 6: Expected Values
Score
Income($)
1300-1430
1300-1430
Total
15000 – 55000
2.22222
2.77777
5
56000 – 96000
1.77777
2.22222
4
Total
4
5
9
Table 6 shows the expected values retrieved by the calculations in table 4

11. The 2 critical value at 5% significance with 1 degree of freedom is 3.841. As
the 2 value is greater than the critical value, 5.760 3.841, the null hypothesis is
rejected and SAT score is assumeddependent from family income.
Discussion/Validity
Limitations
Throughout the investigation between the correlation of SAT scores and
family income, various limitations may have affected the outcome of the results.
One limitationof the data collected could be that it only reflects on the
people who filled in the family income section before signing up for the SAT.
There is no evidence that the data reflects everyone who has taken the SAT
score as there may be people who did not fill that section.
Another limitation could be that not everyone in the world decide to take
the SAT, people who cannot afford it or take alternative tests are being neglected.
Also the data does not confirm of how many SAT takers are being considered.
The data can be proved insufficient and inaccurate for those reasons.
There is also a limitation in the data as it states income of “$100,000 and
above”. That could mean that the data goes on unto family incomes of millions
which is not proportionate to the other ranges of family income given. Due to this
however, that piece of data was left out in the calculations.
Continuing, there might be a limitation to the recording of the data itself as
SAT takers are to take a survey where they mention family income when signing
up for SAT. This might cause a problem as many SAT takers, mostly in ages 1517, do not know the actual income of their family therefore wrong data may be
entered.
Then there could be a limitation to the data due to culture and race. The
data does not mention culture and race which might affect the data as there
might have been more American surveys who mentioned family income
compared to Asian who answered the survey.
Another limitation is that the table of expected values in the
values less than 5 which reduces its validity.
2
test has all

12. Adding on to that, there might be a limitation to the amount of data that
was collected as 9 pieces of data may not prove to be sufficient enough to reflect
the correlation between SAT scores and family income in a world perspective.
Lastly, there may be many other factors taking place when considering the
correlation between SAT scores and family income such as reasons for having a
high family income and IQ of SAT test takers.
Conclusion
Despite of the previously mentioned limitations, the found 2 value, 5.760,
rejects the null hypothesis that SAT scores are independent from family income
and accepts the alternative hypothesis that SAT scores are dependent from
family income. Furthermore,the investigation clearly shows that there is a strong
and positive correlation between SAT score and family income as it can be an
assumed dependence from each other.
Work Cited
Rampell, Catherine. "SAT Scores and Family Income - NYTimes.com." The Economy
and the Economics of Everyday Life - Economix Blog - NYTimes.com. 28 Aug.
2009. Web. 01 Nov. 2010.<http://economix.blogs.nytimes.com/2009/08/27/satscores-and-family-income/>.
Downey, Joel. "SAT Scores Rise with Family Income." Cleveland OH Local News,
Breaking News,Sports & Weather - Cleveland.com. 10 Apr. 2008. Web. 01 Nov.
2010.<http://www.cleveland.com/pdgraphics/index.ssf/2008/04/sat_scores_rise_
with_family_in.html>.
Whiffen, Glen, John Owen, Robert Haese, Sandra Haese, and Mark Bruce. "Two
Variable Statistics." Mathematics for the International Student: Mathematical
Studies SL. By Mal Coad. [S.l.]: Haese And Harris Pub, 2010. 581-82. Print.